24.7 Presentation in the frequency domain

For an absolute summable sequence (xt)t∈ of real numbers and the functions

the inverse of the Fourier transformation exists and the following holds:

Therefore the consideration of sequences in the time domain is equivalent to that in the frequency domain. The Fourier transformation contains the same information as the series itself. In the case of finite time series (xt)t∈T it is sufficient to know the Fourier transformation for finitely many frequency values. A time series with an odd number of observations N = 2 M + 1 can be represented uniquely using the so-called Fourier frequencies

and

implies

(24.10)

For a fuzzy time series (xt*)t∈T such a presentation does not exist in general. The reason for that is the semilinearity of the structure ( ...

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